How to unwrap properly this bevelled mesh (see pic)
Your problem is that the rounded corners of your tiles don't have the same radius as the rounded corners of the texture.  You can work around this by marking the edges of your tiles so you can unwrap each face separately, and then using Unwrap with 'conformal' set.  After you've done that, manually center the top and bottom faces, and shrink them slighly to fit.  You end up with something like this: